Page 57 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.2 Mathematical Formulation of the Algorithm 43

indicates that to evaluate the thermal deformation using mesh 2 in FLAC, it is essen-
tial to translate/transfer the nodal solutions from mesh 1 into mesh 2. Since mesh 1
is totally different from mesh 2, it is only possible to interpolate the nodal solu-
tion of mesh 2, point by point, using the nodal solutions of mesh 1 and the related
elemental information. Thus, for any nodal point in mesh 2, it is necessary to find
out the element, which contains this particular point, in mesh 1. This is the first step,
termed point-searching, in the proposed algorithm. Once a point in mesh 2 is related
to an element (containing this point) in mesh 1, the local coordinates of this point
need to be evaluated using the related information of the element in mesh 1. This
requires an inverse mapping to be carried out for this particular element in mesh 1,
so that the nodal solution of the point in mesh 2 can be consistently interpolated
in the finite element sense. This is the main reason why the proposed algorithm is
called the consistent point-searching interpolation algorithm.
Clearly, for the purpose of developing the consistent point-searching interpola-
tion algorithm, one has to deal with the following two key issues. First, an efficient
searching strategy needs to be developed to limit the number of elements, to which
the inverse mapping is applied, so that the algorithm growth rate can be reduced
to the minimum. Second, the issue of the parametric inverse mapping between the
real (global) and element (local) coordinates should be dealt with in an appropriate
manner. Generally, the inverse mapping problem is a nonlinear one, which requires
numerical solutions for higher order elements. However, for 4-node bilinear quadri-
lateral elements, it is possible to solve this problem analytically. Since most practical
problems in finite element analysis can be modelled reasonably well using 4-node
bilinear quadrilateral elements, the analytical solution to the inverse mapping prob-
lem for this kind of element may find wide applications for many practical problems.
Given the importance of the above two key issues, they are addressed in great detail
in the following sections, respectively.

3.2.1 Point Searching Step

In this section, we deal with the first issue, i.e., the development of an efficient
searching strategy to limit the number of elements, which need to be inversely
mapped. To achieve this, we have to establish a strategy to relate a point in one
mesh to an element in another mesh accurately. Consider a four-node quadrilateral
element shown in Fig. 3.2. If any point (i.e., point A in Fig. 3.2) is located within
this element, then the following equations hold true:
→ → →
12 × 1A = λ 1 k , (3.22)
→ → →
23 × 2A = λ 2 k , (3.23)
→ → →
34 × 3A = λ 3 k , (3.24)
→ → →
41 × 4A = λ 4 k , (3.25)
λ i ≥ 0 (i = 1, 2, 3, 4), (3.26)